Hi Bob,
I have attached a document that outlines the following:
1.  Download and install the driver.
2.  Add the ODBC driver as a data source on the machine.
3.  Connect to the MySQL database in ArcCatalog.
Here are some relevant links:
MySQL ODBC Driver download (5.2.x)
MySQL :: Download Connector/ODBC 
Adding an OLE DB Connection in ArcGIS Desktop.
Adding OLE DB connections to databases—Help | ArcGIS for Desktop 
- Ken